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Since the early work of Gauss and Riemann, differential geometry
has grown into a vast network of ideas and approaches, encompassing
local considerations such as differential invariants and jets as
well as global ideas, such as Morse theory and characteristic
classes. In this volume of the Encyclopaedia, the authors give a
tour of the principal areas and methods of modern differential
geomerty. The book is structured so that the reader may choose
parts of the text to read and still take away a completed picture
of some area of differential geometry. Beginning at the
introductory level with curves in Euclidian space, the sections
become more challenging, arriving finally at the advanced topics
which form the greatest part of the book: transformation groups,
the geometry of differential equations, geometric structures, the
equivalence problem, the geometry of elliptic operators. Several of
the topics are approaches which are now enjoying a resurgence, e.g.
G-structures and contact geometry. As an overview of the major
current methods of differential geometry, EMS 28 is a map of these
different ideas which explains the interesting points at every
stop. The authors' intention is that the reader should gain a new
understanding of geometry from the process of reading this survey.
Since the early work of Gauss and Riemann, differential geometry
has grown into a vast network of ideas and approaches, encompassing
local considerations such as differential invariants and jets as
well as global ideas, such as Morse theory and characteristic
classes. In this volume of the Encyclopaedia, the authors give a
tour of the principal areas and methods of modern differential
geomerty. The book is structured so that the reader may choose
parts of the text to read and still take away a completed picture
of some area of differential geometry. Beginning at the
introductory level with curves in Euclidian space, the sections
become more challenging, arriving finally at the advanced topics
which form the greatest part of the book: transformation groups,
the geometry of differential equations, geometric structures, the
equivalence problem, the geometry of elliptic operators. Several of
the topics are approaches which are now enjoying a resurgence, e.g.
G-structures and contact geometry. As an overview of the major
current methods of differential geometry, EMS 28 is a map of these
different ideas which explains the interesting points at every
stop. The authors' intention is that the reader should gain a new
understanding of geometry from the process of reading this survey.
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